\(\int \frac {x^3}{(c+a^2 c x^2)^2 \arctan (a x)^2} \, dx\) [551]

   Optimal result
   Rubi [N/A]
   Mathematica [N/A]
   Maple [N/A] (verified)
   Fricas [N/A]
   Sympy [N/A]
   Maxima [N/A]
   Giac [N/A]
   Mupad [N/A]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {x}{a^3 c^2 \arctan (a x)}+\frac {x}{a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a^4 c^2}+\frac {\text {Int}\left (\frac {1}{\arctan (a x)},x\right )}{a^3 c^2} \]

[Out]

-x/a^3/c^2/arctan(a*x)+x/a^3/c^2/(a^2*x^2+1)/arctan(a*x)-Ci(2*arctan(a*x))/a^4/c^2+Unintegrable(1/arctan(a*x),
x)/a^3/c^2

Rubi [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]

[In]

Int[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]

[Out]

-(x/(a^3*c^2*ArcTan[a*x])) + x/(a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a^4*c^2) + De
fer[Int][ArcTan[a*x]^(-1), x]/(a^3*c^2)

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx}{a^2}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx}{a^2 c} \\ & = -\frac {x}{a^3 c^2 \arctan (a x)}+\frac {x}{a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{a^3}+\frac {\int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{a}+\frac {\int \frac {1}{\arctan (a x)} \, dx}{a^3 c^2} \\ & = -\frac {x}{a^3 c^2 \arctan (a x)}+\frac {x}{a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a^4 c^2}+\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a^4 c^2}+\frac {\int \frac {1}{\arctan (a x)} \, dx}{a^3 c^2} \\ & = -\frac {x}{a^3 c^2 \arctan (a x)}+\frac {x}{a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c^2}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c^2}+\frac {\int \frac {1}{\arctan (a x)} \, dx}{a^3 c^2} \\ & = -\frac {x}{a^3 c^2 \arctan (a x)}+\frac {x}{a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-2 \frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 a^4 c^2}+\frac {\int \frac {1}{\arctan (a x)} \, dx}{a^3 c^2} \\ & = -\frac {x}{a^3 c^2 \arctan (a x)}+\frac {x}{a^3 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a^4 c^2}+\frac {\int \frac {1}{\arctan (a x)} \, dx}{a^3 c^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]

[In]

Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]

[Out]

Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 8.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}d x\]

[In]

int(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x)

[Out]

int(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^3/((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^2), x)

Sympy [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {x^{3}}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

integrate(x**3/(a**2*c*x**2+c)**2/atan(a*x)**2,x)

[Out]

Integral(x**3/(a**4*x**4*atan(a*x)**2 + 2*a**2*x**2*atan(a*x)**2 + atan(a*x)**2), x)/c**2

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 4.68 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-(x^3 - (a^3*c^2*x^2 + a*c^2)*arctan(a*x)*integrate((a^2*x^4 + 3*x^2)/((a^5*c^2*x^4 + 2*a^3*c^2*x^2 + a*c^2)*a
rctan(a*x)), x))/((a^3*c^2*x^2 + a*c^2)*arctan(a*x))

Giac [N/A]

Not integrable

Time = 120.61 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

int(x^3/(atan(a*x)^2*(c + a^2*c*x^2)^2),x)

[Out]

int(x^3/(atan(a*x)^2*(c + a^2*c*x^2)^2), x)